Trying to gain insight into Nature and considering the physical processes at very small distances, physicists have made attempts to explain the well-known laws of low-energy physics as a consequence of the more fundamental laws of Nature. The contemporary low-energy physics of the electroweak and strong interactions is described by the Standard Model (SM) which uni?es the Glashow-Salam-Weinberg electroweak theory with QCD – the theory of strong interactions. The gauge symmetry group in the SM is : SM G = SU (3)c × SU (2)L × U (1)Y , (1)

which describes elementary particle physics up to the scale ≈ 100 GeV. Recently it was shown that the Family Replicated Gauge Groups (FRGG) of the type SU (n)N × SU (m)N provide new directions for research in high energy physics and quantum ?eld theory. In the Deconstruction of space-time models [1], the authors tried to construct renormalizable asymptotically free 4-dimensional gauge theories which dynamically generate a ?fth dimension (it is possible to obtain more dimensions in this way). Such theories naturally lead to electroweak symmetry breaking, relying neither on supersymmetry nor on strong dynamics at the TeV scale. The new TeV physics is perturbative and radiative corrections to the Higgs mass are ?nite. Thus, we see that the family replicated gauge groups provide a new way to stabilize the Higgs mass in the Standard Model. But there exists quite di?erent way to employ the FRGG.Talk given by L. V. Laperashvili at the Fifth International Conference “Symmetry in Nonlinear Mathematical Physics”, Kyiv, Ukraine, June 23-29, 2003.?

is the fundamental gauge group, which takes right-handed neutrinos and the see-saw mechanism into account. This extended model can describe all modern neutrino experiments, giving a reasonable ?t to all the quark-lepton masses and mixing angles. The gauge group G = Gext contains: 3×8 = 24 gluons, 3×3 = 9 W -bosons, and 3×1+3×1 = 6 Abelian gauge bosons. The gauge group Gext = (SM G × U (1)B?L )3 undergoes spontaneous breakdown (at some orders of magnitude below the Planck scale) to the Standard Model Group SMG which is the diagonal subgroup of the non-Abelian sector of the group Gext . As was shown in Ref. [7], 6 di?erent Higgs ?elds: ω, ρ, W , T , φW S , φB?L break our FRGG-model to the SM. The ?eld φW S corresponds to the Weinberg-Salam Higgs ?eld of Electroweak theory. Its vacuum expectation value (VEV) is ?xed by the Fermi constant: φW S = 246 GeV, so that we have only 5 free parameters – ?ve VEVs: ω , ρ , W , T , φB?L to ?t the experiment in the framework of the SM. These ?ve adjustable parameters were used with the aim of ?nding the best ?t to experimental data for all fermion masses and mixing angles in the SM, and also to explain the neutrino oscillation experiments. Experimental results on solar neutrino and atmospheric neutrino oscillations from Sudbury Neutrino Observatory (SNO Collaboration) and the Super-Kamiokande Collaboration have been used to extract the following parameters:2 2 ?m2 solar = m2 ? m1 ,

?m2 = m2 ? m2 , atm 3 2

tan2 θsolar = tan2 θ12 ,

tan2 θatm = tan2 θ23 (7)

where m1 , m2 , m3 are the hierarchical left-handed neutrino e?ective masses for the three families. We also use the CHOOZ reactor results. It is assumed that the fundamental Yukawa couplings in our model are of order unity and so we make order of magnitude predictions. The typical ?t is shown in Table I. As we can see, the 5 parameter order of magnitude ?t is encouraging.

Family Replicated Gauge Group Models

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Table 1. Best ?t to conventional experimental data. All masses are running masses at 1 GeV except the top quark mass which is the pole mass.

– for right-handed (heavy) neutrinos. Finally, we conclude that our theory with the FRGG-symmetry is very successful in describing experiment. The best ?t gave the following values for the VEVs: W ≈ 0.157, T ≈ 0.077, ω ≈ 0.244, ρ ≈ 0.265 (10)

in the “fundamental units”, MP l = 1, and φB?L ≈ 5.25 × 1015 GeV (11)

which gives the see-saw scale: the scale of breakdown of the U (1)B?L groups (? 5 × 1015 GeV).

3

The Problem of Monopoles in the Standard and Family Replicated Models

The aim of the present Section is to show, following Ref. [8], that monopoles cannot be seen in the Standard Model and in its usual extensions in the literature up to the Planck scale:

4

C. D. Froggatt, L. V. Laperashvili, H. B. Nielsen, Y. Takanishi

MP l = 1.22 × 1019 GeV, because they have a huge magnetic charge and are completely con?ned or screened. Supersymmetry does not help to see monopoles. In theories with the FRGG-symmetry the charge of monopoles is essentially diminished. Then monopoles can appear near the Planck scale and change the evolution of the ?ne structure constants αi (t) (here i = 1, 2, 3 corresponds to U (1), SU (2) and SU (3)), t = log(?2 /?2 ), where R ? is the energy variable and ?R is the renormalisation point. Let us consider the “electric” and “magnetic” ?ne structure constants: α= g2 4π and α= ? g2 ? , 4π (12)

where α and αs are the electromagnetic and SU (3) ?ne structure constants respectively, Y is the weak hypercharge, and θMS is the Weinberg weak angle in MS scheme. Using RGEs with experimentally established parameters, it is possible to extrapolate the experimental values of the three inverse running constants α?1 (?) from the Electroweak scale to the Planck scale (see i Fig. 1). In this connection, it is very attractive to include gravity. The quantity: αg = ? ?P l2

(17)

plays the role of the running ”gravitational ?ne structure constant” (see Ref. [8]) and the evolution of its inverse is presented in Fig. 1 together with the evolutions of α?1 (?). i Assuming the existence of the Dirac relation: g? = 2π for minimal charges, we have the g following expression for the renormalised charges g and g [9]: ? 1 α(t)? (t) = . α 4 (18)

Family Replicated Gauge Group Models

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70 60 50

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40 30 20 10 0 0 2

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4

6

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10??

12

14??

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Figure 1.

Using the Dirac relation, it is easy to estimate (in the simple SM) the Planck scale value of α(?P l ) (minimal for U (1)Y gauge group): ? 5 α(?P l ) = α?1 (?P l )/4 ≈ 55.5/4 ≈ 14 . ? (19) 3 1 This value is really very big compared with the estimate (15) and, of course, with the critical coupling αcrit ≈ 1, corresponding to the con?nement – decon?nement phase transition in the ? lattice U (1) gauge theory. Clearly we cannot make a perturbation approximation with such a strong coupling α. It is hard for such monopoles not to be con?ned. ? There is an interesting way out of this problem if one wants to have the existence of monopoles, namely to extend the SM gauge group so cleverly that certain selected linear combinations of charges get bigger electric couplings than the corresponding SM couplings. That could make the monopoles which, for these certain linear combinations of charges, couple more weakly and thus have a better chance of being allowed “to exist”. An example of such an extension of the SM that can impose the possibility of allowing the existence of free monopoles is just Family Replicated Gauge Group Model (FRGGM). FRGGs of type [SU (N )]Nf am lead to the lowering of the magnetic charge of the monopole belonging to one family: αone f amily = ? α ? Nf am .(2,3)

1 where N ? = 2 Nf am (Nf am + 1). For Nf am = 3 and [U (1)]3 , we have: αone family = α(1) /6 (six ? ? times smaller!). This result was obtained previously in Ref. [10]. According to the FRGGM, at some point ? = ?G < ?P l (or really in a couple of steps) the fundamental group G ≡ Gext undergoes spontaneous breakdown to its diagonal subgroup:

G ?→ Gdiag.subgr. = {g, g, g||g ∈ SM G},

(22)

which is identi?ed with the usual (low-energy) group SMG. In the Anti-GUT-model [2, 3] the FRGG breakdown was considered at ?G ? 1018 GeV. But the aim of this investigation is to show that we can see quite di?erent consequences of the extension of the SM to FRGGM, if the G-group undergoes the breakdown to its diagonal subgroup (that is, SMG) not at ?G ? 1018 GeV, but at ?G ? 1014 or 1015 GeV, i.e. before the intersection of α?1 (?) with α?1 (?) at ? ≈ 1016 GeV. In this case, in the region ?G < ? < ?P l 2 3 there are three SM G × U (1)B?L groups for the three FRGG families, and we have a lot of fermions, mass protected or not mass protected, belonging to usual families or to mirror ones. In the FRGGM the additional 5 Higgs bosons, with their large VEVs, are responsible for the mass protection of a lot of new fermions appearing in the region ? > ?G . Here we denote the total number of fermions NF , which is di?erent to Nf am . Also the role of monopoles can be important in the vicinity of the Planck scale: they give contributions to the beta-functions and change the evolution of the α?1 (?). Finally, we obtain the following RGEs: N d(α?1 (?)) bi i = + M β (m) (? U (1) ) α dt 4π αi where bi are given by the following values: bi = (b1 , b2 , b3 ) 1 4NF ? NS , = (? 3 10 22 4NF 1 NV ? ? NS , 3 3 6 4NF ). 3(i)

(23)

11NV ?

(24)

The integers NF , NS , NV , NM are respectively the total numbers of fermions, Higgs bosons, vector gauge ?elds and scalar monopoles in the FRGGM considered in our theory. In our FRGG model we have NV = 3, because we have 3 times more gauge ?elds (Nf am = 3), in comparison with the SM and one Higgs scalar monopole in each family. We have obtained the evolutions of α?1 (?) near the Planck scale by numerical calculations i (2,3) (1) for: ?G = 1014 GeV, NF = 18, NS = 6, NM = 6, NM = 3. Fig. 2 shows the existence of the uni?cation point. We see that in the region ? > ?G a lot of new fermions, and a number of monopoles near the Planck scale, change the one-loop approximation behaviour of α?1 (?) which we had in the i SM. In the vicinity of the Planck scale these evolutions begin to decrease, as the Planck scale ? = ?P l is approached, implying the suppression of asymptotic freedom in the non-Abelian theories. Fig. 2 gives the following Planck scale values for the αi : α?1 (?P l ) ≈ 13 1 α?1 (?P l ) ≈ 19 2 α?1 (?P l ) ≈ 24 . 3 (25)

Fig. 2 demonstrates the uni?cation of all gauge interactions, including gravity (the intersection of α?1 with α?1 ), at g i α?1 T ≈ 27 GU and xGU T ≈ 18.4 . (26)

Here we can expect the existence of [SU (5)]3 or [SO(10)]3 (SUSY or not SUSY) uni?cation.

Family Replicated Gauge Group Models

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50

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40

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Figure 2.

Considering the predictions of such a theory for low-energy physics and cosmology, maybe in future we shall be able to answer the question: Does the uni?cation [SU (5)]3 or [SO(10)]3 really exist near the Planck scale? Recently F. S. Ling and P. Ramond [11] considered the group of symmetry [SO(10)]3 and showed that it explains the observed hierarchies of fermion masses and mixings.